In mathematics, and in particular algebraic geometry, K-stability is an algebro-geometric stability condition for projective algebraic varieties and complex manifolds. K-stability is of particular importance for the case of Fano varieties, where it is the correct stability condition to allow the formation of moduli spaces, and where it precisely characterises the existence of Kähler–Einstein metrics. The first attempt to define K-stability for Fano manifolds was made by Gang Tian in 1997, in response to a conjecture of Shing-Tung Yau from 1993 that there should exist a stability condition which characterises the existence of a Kähler–Einstein metric on a Fano manifold. It was defined in reference to the K-energy functional previously introduced by Toshiki Mabuchi. Tian's definition of K-stability was later replaced by a purely algebro-geometric refinement that was first formulated by Simon Donaldson in 2001. K-stability has become an important notion in the study and classification of Fano varieties. In 2012 Xiuxiong Chen, Donaldson, and Song Sun proved that a smooth Fano manifold is K-polystable if and only if it admits a Kähler–Einstein metric. (Tian then announced a nearly identical proof, under circumstances that resulted in a bitter priority dispute.) This theorem was later generalised to singular K-polystable Fano varieties due to the work of Berman–Boucksom–Jonsson, Li and Liu-Xu-Zhuang. K-stability is important in constructing moduli spaces of Fano varieties, where observations going back to the original development of geometric invariant theory show that it is necessary to restrict to a class of stable objects to form good moduli. It is now known through the work of Chenyang Xu and others that there exists a projective good moduli space of K-polystable Fano varieties. Due to the reformulations of the K-stability condition by Fujita–Li, the K-stability of Fano varieties may be explicitly computed in practice. Which Fano varieties are K-stable is well understood in dimension one, two, and three.
Definition and characterisations The notion of K-stability for Fano manifolds was originally specified using differential geometry by Tian, who extended the purely analytical notion of the Futaki invariant of a vector field to the case of certain normal varieties with orbifold singularities. This was later reformulated in a purely algebro-geometric form by Donaldson, but this general definition lost a direct link to the geometry of Fano varieties, instead making sense for the broader class of all projective varieties. Work of Tian shows that the Donaldson–Futaki invariant specifying the weight of the C ∗ {\displaystyle \mathbb {C} ^{*}} -action on the central fibre of a test configuration can be computed in terms of certain intersection numbers (corresponding to the weight of an action on the so-called CM line bundle). In the Fano case these intersection numbers, which involve the anticanonical divisor of the variety and its test configuration, can be given powerful alternative characterisations in terms of the algebraic and birational geometry of the Fano variety. Thus in the case of Fano varieties, there are many different but equivalent characterisations of K-stability, and some of these characterisations lend themselves to explicit calculation or easier proofs of results. In this section all definitions are stated in the generality of a Q {\displaystyle \mathbb {Q} } -Fano variety, which is a Fano variety with ample Q {\displaystyle \mathbb {Q} } -Cartier anticanonical divisor and at worst Kawamata log terminal (klt) singularities. The definitions of K-stability can be made for any Q {\displaystyle \mathbb {Q} } -Gorenstein Fano variety (that is, any Fano variety where the anticanonical divisor is Q {\displaystyle \mathbb {Q} } -Cartier), however it was proven by Odaka that every K-semistable Fano variety has at worst klt singularities, so for the purpose of studying K-stability it suffices to assume at worst klt singularities. Every definition can be extended in a straightforward way to Q {\displaystyle \mathbb {Q} } -log Fano pairs, a pair ( X , Δ ) {\displaystyle (X,\Delta )} of a klt variety X and klt divisor such that − ( K X + Δ ) {\displaystyle -(K_{X}+\Delta )} is ample and Q {\displaystyle \mathbb {Q} } -Cartier.
Traditional definition
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